Pertanyaan

Jika adalah turunan pertama fungsi x dan adalah turunan keduanya, maka tentukan turunan kedua dari fungsi f ( x ) = x + 1 2 x .

Jika $fraction numerator d over denominator d x end fraction open square brackets f left parenthesis x right parenthesis close square brackets equals f apostrophe left parenthesis x right parenthesis$ adalah turunan pertama fungsi $x$ dan $fraction numerator d over denominator d x end fraction open square brackets f left parenthesis x right parenthesis close square brackets equals f double apostrophe left parenthesis x right parenthesis$ adalah turunan keduanya, maka tentukan turunan kedua dari fungsi $f (x) = \frac{2 x}{x + 1}$ .

R. Bella

Master Teacher

Mahasiswa/Alumni UIN Syarif Hidayatullah Jakarta

Jawaban terverifikasi

Jawaban

turunan kedua dari fungsi adalah .

turunan kedua dari fungsi $f left parenthesis x right parenthesis equals fraction numerator 2 x over denominator x plus 1 end fraction$ adalah $table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell f double apostrophe left parenthesis x right parenthesis end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table fraction numerator negative 4 x minus 4 over denominator open parentheses x plus 1 close parentheses to the power of 4 end fraction$ .

Pembahasan

Ingat bahwa: Dengan dan . Dengan dan . Jadi turunan kedua dari fungsi adalah .

Ingat bahwa:

$f left parenthesis x right parenthesis equals u over v rightwards arrow f left parenthesis x right parenthesis equals fraction numerator u apostrophe times v minus u times v apostrophe over denominator v squared end fraction$

Dengan u equals 2 x rightwards arrow u apostrophe equals 2 dan v equals x plus 1 rightwards arrow v apostrophe equals 1 .

$table attributes columnalign right center left columnspacing 0px end attributes row cell f left parenthesis x right parenthesis end cell equals cell fraction numerator 2 x over denominator x plus 1 end fraction end cell row cell f apostrophe left parenthesis x right parenthesis end cell equals cell fraction numerator u apostrophe times v minus u times v apostrophe over denominator v squared end fraction end cell row blank equals cell fraction numerator 2 times open parentheses x plus 1 close parentheses minus 2 x times 1 over denominator open parentheses x plus 1 close parentheses squared end fraction end cell row blank equals cell fraction numerator 2 x plus 2 minus 2 x over denominator open parentheses x plus 1 close parentheses squared end fraction end cell row cell f apostrophe left parenthesis x right parenthesis end cell equals cell fraction numerator 2 over denominator x squared plus 2 x plus 1 end fraction end cell end table$

Dengan u equals 2 rightwards arrow u apostrophe equals 0 dan v equals x squared plus 2 x plus 1 rightwards arrow v apostrophe equals 2 x plus 2 .

$table attributes columnalign right center left columnspacing 0px end attributes row cell f apostrophe left parenthesis x right parenthesis end cell equals cell fraction numerator 2 over denominator x squared plus 2 x plus 1 end fraction end cell row cell f double apostrophe left parenthesis x right parenthesis end cell equals cell fraction numerator u apostrophe times v minus u times v apostrophe over denominator v squared end fraction end cell row cell f double apostrophe left parenthesis x right parenthesis end cell equals cell fraction numerator 0 times open parentheses x squared plus 2 x plus 1 close parentheses minus 2 open parentheses 2 x plus 2 close parentheses over denominator open parentheses x squared plus 2 x plus 1 close parentheses squared end fraction end cell row blank equals cell fraction numerator negative 4 x minus 4 over denominator open parentheses open parentheses x plus 1 close parentheses squared close parentheses squared end fraction end cell row cell f double apostrophe left parenthesis x right parenthesis end cell equals cell fraction numerator negative 4 x minus 4 over denominator open parentheses x plus 1 close parentheses to the power of 4 end fraction end cell end table$

Jadi turunan kedua dari fungsi $f left parenthesis x right parenthesis equals fraction numerator 2 x over denominator x plus 1 end fraction$ adalah $table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell f double apostrophe left parenthesis x right parenthesis end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table fraction numerator negative 4 x minus 4 over denominator open parentheses x plus 1 close parentheses to the power of 4 end fraction$ .